The terrorist group has mostly been populating the eastern province of Nangarhar, from where it has carried out high-profile brutal attacks at major population centers across the country.
Organisers were under pressure to avoid last weekend's debacle that forced the Independent Election Commission (IEC) to extend the nationwide ballot by a day.
People in Kandahar Province went to the polls on Saturday, while elections have yet to be held in central Ghazni Province, which is still reeling from the Taliban's takeover in August. But hiccups with biometric devices and voter lists persisted. At some sites the gadgets were not used at all.
"I have defied all the threats of attacks and explosions to vote".
The attack took place after a meeting with the top USA commander in Afghanistan, General Scott Miller.More news: Gaza rockets hit Israel after five Palestinians killed in border flareup
No group claimed responsibility for the attack. According to it, almost four million people voted last weekend, in spite of many attacks perpetrated against the voters.
Many suspect a significant number of those on the voter roll were based on fake identification documents that fraudsters planned to use to stuff ballot boxes.
The ballot, which the Taliban had vowed to attack, was marred by deadly violence, with hundreds killed or wounded in scores of attacks.
According to the IEC, 111 candidates from Kandahar were competing for 11 seats, including three for women, at the Lower House of the Afghan Parliament.
The election, held up for years amid disputes between political factions over voting rules, is one of the biggest tests of Afghanistan's ability to protect itself since an global security force mostly withdrew in 2014.
Preliminary results of the elections held on October 20-21 are expected to be announced on November 10, while final results are due by December 20.